Theorem e2ebindALT 44949

Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in e2ebindVD 44932. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
e2ebindALT	⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Proof of Theorem e2ebindALT

Step	Hyp	Ref	Expression
1		axc11n 2431	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2		nfe1 2150	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
3	2	19.9 2205	. . 3 ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
4		excom 2162	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
5		nfa1 2151	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 𝑦 = 𝑥
6		id 22	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)
7		biid 261	. . . . . . . . 9 ⊢ (𝜑 ↔ 𝜑)
8	7	a1i 11	. . . . . . . 8 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
9	8	drex1 2446	. . . . . . 7 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
10	6, 9	syl 17	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
11	5, 10	alrimi 2213	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → ∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑))
12		exbi 1847	. . . . 5 ⊢ (∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑) → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑))
13	11, 12	syl 17	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑))
14		bitr 805	. . . . . 6 ⊢ (((∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) ∧ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)) → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑))
15	14	ex 412	. . . . 5 ⊢ ((∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑) → ((∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)))
16	15	impcom 407	. . . 4 ⊢ (((∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) ∧ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)) → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑))
17	4, 13, 16	sylancr 587	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑))
18		bitr3 352	. . . 4 ⊢ ((∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑) → ((∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)))
19	18	impcom 407	. . 3 ⊢ (((∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) ∧ (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)) → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))
20	3, 17, 19	sylancr 587	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))
21	1, 20	syl 17	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784

This theorem is referenced by: (None)